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## Quasi-Feynman formulas for the Schrödinger equation on compact and non-compact manifolds

math.
arXiv.
Cornell University
,
2021.

Florido Calvo F. A., Remizov I.

Dynamics of closed quantum systems on curves, surfaces and more general manifolds is governed by the Schroedinger equation with time-independent Hamiltonian. Solving Cauchy problem for this equation provides full information on the future and the past of the system if we know the state of the system at the initial moment of time t=0. However, in interesting cases the Schroedinger equation is a partial derivative differential equation with variable coefficients, hence it is not easy to obtain the explicit solution. There exist only few papers that are devoted to this topic, and most of them are devoted to the case of compact manifolds, representing solution in the form of a Feynman formula with the help of the Chernoff product formula for operator semigroups. In the present paper we extend this approach to the class of manifolds of bounded geometry, which includes all compact manifolds and a wide class of non-compact manifolds. The solution is given as a limit of a sequence of functions on the manifold that converge in mean square.

Remizov I., Potential analysis 2020 Vol. 52 P. 339-370

In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients ...

Added: September 30, 2018

Anikin A., Brüning J., Dobrokhotov S. et al., Russian Journal of Mathematical Physics 2019 Vol. 26 No. 3 P. 265-276

In this paper, we consider the spectral problem for the magnetic Schrödinger operator on the 2-D plane (x1, x2) with the constant magnetic field normal to this plane and with the potential V having the form of a harmonic oscillator in the direction x1 and periodic with respect to variable x2. Such a potential can ...

Added: September 18, 2019

Karasev M., Vybornyi E., Journal of Mathematical Physics 2016

We consider the one-dimensional Schrodinger operator in the semiclassical regime assuming that its double-well potential is the sum of a finite "physically given" well and a square shape probing well whose width or depth can be varied (tuned). We study the dynamics of an initial state localized in the physical well. It is shown that ...

Added: October 23, 2015

Vybornyi E., Theoretical and Mathematical Physics 2014 Vol. 181 No. 2 P. 1418-1427

We propose an operator method for calculating the semiclassical asymptotic form of the energy splitting value in the general case of tunneling between symmetric orbits in the phase space. We use this approach in the case of a particle on a circle to obtain the asymptotic form of the energy tunneling splitting related to the ...

Added: August 5, 2014

Vedenin A., Galkin V., Karatetskaia E. et al., Speed of convergence of Chernoff approximations to solutions of evolution equations / Cornell University. Series arXiv "math". 2020.

This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations. ...

Added: October 12, 2019

Vybornyi E., Наноструктуры. Математическая физика и моделирование 2015 Т. 12 № 1 С. 5-84

We consider the problem of constructing semiclassical asymptotic expansions of discrete spectrum and the corresponding stationary states of one-dimensional Schrödinger operator in the case of resonance tunneling. We consider two basic models: tunneling in an asymmetric double-well potential on a line and momentum tunneling of a particle in a potential field on a circle. For ...

Added: February 12, 2016

Galkin O., Galkina S., Some parabolic equations for measures and Gaussian semigroups / Cornell University. Series math "arxiv.org". 2020. No. arXiv:2012.07174.

This short communication (preprint) is devoted to mathematical study of evolution equations that are important for mathematical physics and quantum theory; we present new explicit formulas for solutions of these equations and discuss their properties. The results are given without proofs but the proofs will appear in the longer text which is now under preparation.
In ...

Added: December 13, 2020

Vedenin A., Воеводкин В. С., Galkin V. et al., Математические заметки 2020 Т. 108 № 3 С. 463-468

This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations. ...

Added: October 21, 2019

Dragunova K., Гаращенкова А. А., Remizov I., Numerical Study of the Rate of Convergence of Chernoff Approximations to Solutions of the Heat Equation / Cornell University. Series arXiv "math". 2021.

Chernoff approximations are a flexible and powerful tool of functional analysis, which can be used, in particular, to find numerically approximate solutions of some differential equations with variable coefficients. For many classes of equations such approximations have already been constructed, however, the speed of their convergence to the exact solution has not been properly studied. ...

Added: December 16, 2021

Spiridonov V., Self-similar potentials in quantum mechanics and coherent states / Cornell University Library. 2020.

A brief description of the relations between the factorization method in quantum mechanics, self-similar potentials, integrable systems and the theory of special functions is given. New coherent states of the harmonic oscillator related to the Fourier transformation are constructed. ...

Added: September 9, 2020

Bruning J., Grushin V. V., Dobrokhotov S. Y., Математические заметки 2012 Т. 92 № 2 С. 163-180

An example of Schrodinger and Klein-Gordon equations with fast oscillating coefficients is used to show that they can be averaged by an adiabatic approximation based on V.P. Maslov's operator method. ...

Added: December 24, 2012

Vybornyi E., Наноструктуры. Математическая физика и моделирование 2015 Т. 13 № 2 С. 43-54

We conceder semiclassical asymptotics of the energy levels shift of the Schrödinger operator discrete spectrum with a one-dimensional single-well potential that appears due to a deformation of the potential in the classically forbidden region. Since such a deformation of the potential effects on the quantum particle only due to the tunneling effects, then the corresponding ...

Added: February 18, 2016

Chernyshev V. L., Russian Journal of Mathematical Physics 2016 Vol. 23 No. 3 P. 348-354

On a two-dimensional surface, a Schrödinger operator is considered with a potential whose critical points form a closed curve. We pose the problem of describing the semiclassical spectral series corresponding to this curve. The standard construction for describing the spectral series corresponding to isolated nondegenerate equilibria or to periodic trajectories of Hamiltonian systems is not ...

Added: October 25, 2014

Bershtein M., Feigin B. L., Merzon G., Selecta Mathematica, New Series 2018 Vol. 24 No. 1 P. 21-62

We study plane partitions satisfying condition a_{n+1,m+1}=0 (this condition is called “pit”) and asymptotic conditions along three coordinate axes. We find the formulas for generating function of such plane partitions. Such plane partitions label the basis vectors in certain representations of quantum toroidal gl1 algebra, therefore our formulas can be interpreted as the characters of ...

Added: October 24, 2018

Popov V., Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Danilov B. R., Moscow University Computational Mathematics and Cybernetics 2013 Vol. 37 No. 4 P. 180-188

The article investigates a model of delays in a network of functional elements (a gate network) in an arbitrary finite complete basis B, where basis elements delays are arbitrary positive real numbers that are specified for each input and each set of boolean variables supplied on the other inputs. Asymptotic bounds of the form τ ...

Added: December 2, 2019

Okounkov A., Aganagic M., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 565-600

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties.
This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik Zamolodchikov and dynamical q-difference equations. ...

Added: October 25, 2018

Красноярск: ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

Фонарева А. В., Gaydukov R., Russian Journal of Mathematical Physics 2021 Vol. 28 No. 2 P. 224-243

A subsonic flow of a viscous compressible fluid in a two-dimensional channel with small periodic or localized irregularities on the walls for large Reynolds numbers is considered. A formal asymptotic solution with double-deck structure of the boundary layer is constructed. A nontrivial time hierarchy is discovered in the decks. An analysis of the scales of irregularities at ...

Added: March 22, 2021

Levashov M., Кухаренко А. В., Вопросы защиты информации 2018 № 2 С. 66-71

Рассматривается статистическая модель одного этапа системы фрод-мониторинга транзакций в интернет-банкинге. Построен и рассчитан близкий к отношению правдоподобия критерий отсева мошеннических транзакций. Для выборочных распределений, полученных на выборке объема в 1 млн реальных транзакций, вычислены параметры эффективности этого критерия. ...

Added: June 14, 2018

Min N., Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Vyalyi M., Дискретная математика 1991 Т. 3 № 3 С. 35-45

Added: October 17, 2014